The product of all solutions of the equation , is: [2025]

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Solution

Q.
The product of all solutions of the equation $e^{5 {(\log_{e} x)}^{2} + 3} = x^{8}, x > 0$ , is: [2025]

1 $e^{2}$

2 e

3 $e^{8 / 5}$

4 $e^{6 / 5}$

Ans.
(3)

We have, $e^{5 {(\log_{e} x)}^{2} + 3} = x^{8}, x > 0$

Taking log on both sides, we get

$5 {(\ln x)}^{2} + 3 = 8 \ln x \Rightarrow 5 {(\ln x)}^{2} + 3 - 8 \ln x = 0$

Put ln x = t

$\Rightarrow 5 t^{2} - 8 t + 3 = 0 \Rightarrow t_{1} + t_{2} = \frac{8}{5}$

$∴ \ln x_{1} + \ln x_{2} = 8 / 5 \Rightarrow \ln (x_{1} x_{2}) = 8 / 5 \Rightarrow x_{1} x_{2} = e^{8 / 5}$ .

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